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Harshad n.  and  Proth n.

First numbers that belong to both sets:

3, 5, 9, 81, 209, 225, 481, 513, 801, 1729, 2241, 2625, 3969, 4225, 5121, 5505, 9345, 11265, 12033, 12673, 17153, 17409, 19201, 33537, 34305, 34561, 38657, 41985, 50945, 51201, 55809, 56321, 69633, 81921, 83457, 97281, 103425, 103937, 111105, 114177, 120321, 124929, 138753, 146433, 152577, 167425, 167937, 173569, 194049, 207873, 209921, 210945, 221697, 249345, 253441, 263169, 282625, 290817, 300033, 318465, 370689, 372737, 373761, 404481, 429057, 456705, 484353, 512001, 533505, 610305, 622593, 674817, 691201, 705537, 727041, 733185, 743425, 754689, 760833, 776193, 784385, 792577, 816129, 871425, 874497, 882689, 907265, 926721, 982017, 1009665, 1025025, 1041409, 1064961, 1071105, 1153025, 1175553, 1181697, 1216513, 1230849, 1243137, 1286145, 1372161, 1384449, 1441793, 1501185, 1507329, 1562625, 1617921, 1630209, 1665025, 1673217, 1728513, 1839105, 1894401, 1976321, 1992705, 2004993, 2017281, 2060289, 2115585, 2117633, 2125825, 2146305, 2170881, 2195457, 2226177, 2281473, 2316289, 2392065, 2398209, 2404353, 2406401, 2447361, 2502657, 2510849, 2613249, 2662401, 2719745, 2723841, 2834433, 2838529, 2945025, 3000321, 3045377, 3047425, 3055617, 3166209, 3276801, 3307521, 3332097, 3364865, 3411969, 3459073, 3497985, 3508225, 3553281, 3817473, 3940353, 3958785, 3989505, 4009985, 4020225, 4050945, 4161537, 4222977, 4272129, 4382721, 4468737, 4493313, 4603905, 4628481, 5033985, 5046273, 5120001, 5173249, 5509121, 5566465, 5718017, 5812225, 5820417, 5931009, 6017025, 6152193, 6250497, 6385665, 6533121, 6590465, 6705153, 7147521, 7258113, 7467009, 7684097, 7700481, 7811073, 7872513, 7880705, 8032257, 8253441, 8278017, 8364033, 8613889, 8658945, 8683521, 8781825, 8806401, 9089025, 9113601, 9170945, 9740289, 9764865, 9912321, 9949185, 10010625, 10113025, 10244097, 10321921, 10403841, 10465281, 10698753, 11022337, 11063297, 11128833, 11177985, 11239425, 11341825, 11436033, 11444225, 11460609, 11927553, 12013569, 12029953, 12120065, 12210177, 12234753, 12263425, 12345345, 12677121, 12738561, 13172737, 13185025, 13340673, 13451265, 13500417, 13729793, 13783041, 13824001, 13955073, 14000129, 14016513, 14106625, 14114817, 14225409, 14270465, 14360577, 14766081, 14954497, 15028225, 15171585, 15220737, 15331329, 15339521, 15441921, 15458305, 15478785, 15552513, 15663105, 16105473, 16216065, 16449537, 16478209, 16539649, 16601089, 16793601, 17211393, 17604609, 17842177, 17874945, 18227201, 18317313, 18366465.

Some larger examples :